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Tropical linear algebra and its applications in optimisationучебный курс
- Автор: Гутерман А.Э.
- Год создания: 2010
- Организация: МГУ имени М.В. Ломоносова
- Курс на иностранном языке
- Описание: Tropical algebra (sometimes called max or min algebra) is a set of real numbers equipped with the maximum operation instead of usual addition and addition instead of usual multiplication. Under these operations this is an algebraic structure called a semiring. Such structures naturally appear in modern scheduling theory, control theory, optimization, dynamical systems and networks. Tropical arithmetic allows to reduce difficult non-linear problems to the linear problems but over tropical semiring. Therefore, to investigate these problems it is necessary to develop linear algebra in the tropical case. This subject is very actual nowdays and it will be a main topic of our considerations. We will give particular emphasize to methods of solution for tropical systems, spectral theory of tropical matrices, and applications to sсhedulling theory. Program: Tropical algebra. Basic concepts and definitions. Vectors and matrices. Tropical matrices and graphs. Tropical models. Scheduling problem, periodic schedules. Solutions of tropical linear systems. Explicit formula for one solution. Eigenvalues and eigenvectors. Eigenvalues and the maximal average cyclic weight of the graph. Karp’s algorithm. Numerical procedures for eigenvalues of irreducible matrices. Howard’s algorithm. Numerical procedures for eigenvalues of reducible matrices. Eigenvectors. Algorithms to find the eigenvectors. Linear dependence over tropical semiring. Gondran-Minoux, weak and strong linear independence and their relations. The characterization of the eigenspace. Cyclicity and transient time. Tropical methods in synchronization. Tropical models for railway systems. The course is delivered in English. The course was delivered: 1. Moscow State University, Moscow-Rome school on Matrix Methods, October 2010, 2. University of Lisboa, Faculty of Science, Centro de Estruturas Lineares e Combinatórias, http://celc.ciencias.ulisboa.pt/meetings/guterman-kreinesMay2012.html, May-June 2012, 3. Memorial University of Newfoundland, Atlantic Algebra Centre, http://www.mun.ca/aac/guterman.pdf, October 2012.
- Добавил в систему: Гутерман Александр Эмилевич
Преподавание курса
- 1 октября 2010 - 14 октября 2010 Гутерман Александр Эмилевич
- МГУ имени М.В. Ломоносова, Механико-математический факультет, Отделение математики, Кафедра высшей алгебры
- факультативная, лекции, 24 часов
- Автор: Гутерман А.Э.